Deck 12: What Is Algebra

Question 1

Express symbolically, using variables, the general properties in algebra that are illustrated.

A) $( 400 + 25 ) \times 3 = ( 400 \times 3 ) + ( 25 \times 3 )$
B) $( 57 \times 16 ) + ( 43 \times 16 ) = ( 57 + 43 ) \times 16$
C) $( 600 + 32 ) \div 4 = ( 600 \div 4 ) + ( 32 \div 4 )$
D) $\frac { 75 + 18 } { 3 } = \frac { 75 } { 3 } + \frac { 18 } { 3 }$

Answer 1

A) $( a + b ) \times c = ( a \times c ) + ( b \times c )$ , or $( a + b ) c = ( a c ) + ( b c )$ , or just ac + bc
B) $( a \times b ) + ( c \times b ) = ( a + c ) \times b$
C) $( a + b ) \div c = ( a \div c ) + ( b \div c )$
D) $\frac { a + b } { c } = \frac { a } { c } + \frac { b } { c }$ ^. Compare this answer with (a + b) ÷ c = (a ÷ c) + (b ÷ c).

Question 2

Name the property (or properties) that justifies each of the following.

A) (700 + 60 + 3) + (200 + 30 + 5) = (3 + 5) + (60 + 30) + (700 + 200)
B) 5 × (17 + 3) = (5 × 17) + (5 × 3)
C) 984 + (717 + 563) = 984 + (563 + 717)
D) (56 × 89) ×

= 56 × 89
E)
(Hint: More than one property!)
F)
G)
H) 5 × (17 × 3) = 5 × (3 × 17)

Answer 2

A) commutativity and associativity of addition
B) distributivity (of multiplication over addition)
C) commutativity of addition (not associativity)
D) One is the multiplicative identity.
E) commutativity and associativity of addition; additive inverse property; zero is the additive identity
F) distributivity (of multiplication over addition)
G) One is the multiplicative identity.
H) commutativity of multiplication (not associativity)

Question 3

Test each algebra statement to see whether they appear to be true in general. If a statement appears always to be true, draw a diagram to justify the statement. Add an explanation if the diagram is not self-explanatory. If a statement is not true in general, give a counterexample.

A)
B)
C)
D)
E)

Answer 3

A) Not true in general; student should have shown a counterexample.
B) True. Sample diagram:
C) Not true in general; student should have shown a counterexample.
D) Not true in general; student should have shown a counterexample.
E) Not true in general; student should have shown a counterexample.

Question 4

Make a drawing to justify
.

Question 5

The "balance" diagram below shows x + 2 = x³. (If it is not true, give a correct equation.)

Question 6

Make a drawing of a balance for the equation
. Solve the equation by showing actions with the balance, one step at a time.

Question 7

Show your mastery of the conventional order of operations by evaluating each.

A) 6 + 3 × 7 - (2 + ^-1) ⁵
B) 10 - 4 ÷ 3 × 2 + 1
C) 10 - 4 ÷ 3 × (2 + 1)
D) $4 x ^ { 2 } - 7 x + {}^{-} 2$ , when $x = 3$
E) $9 - 3 x - 5 x ^ { 3 }$ , when $x = {}^{-} 2$

Question 8

In each part, use the given, correct algebraic statements to answer the calculations.

A) Part I:
$( x + 4 ) ( 2 x + 1 ) = 2 x ^ { 2 } + 9 x + 4$
implies that 14_seventeen × 21_seventeen = ______seventeen.
B) Part II:
$( 2 x + 3 ) ( 2 x + 1 ) = 4 x ^ { 2 } + 8 x + 3$
implies that 23_eleven × 21_eleven = ______eleven.
C) Bonus: What other bases could be used in parts I and II?

Question 9

Complete the following and tell how they are alike.

A) $\frac { 2 } { 3 } + \frac { 7 } { 12 }$
B) $\frac { 2 } { x + 2 } + \frac { 3 x + 4 } { ( x + 2 ) ( x + 3 ) }$

Question 10

A) Part I: It is correct that 12_nine × 32_nine = 384_nine. How might that inform ?
B) Part II: Explain the "might" in part A by considering 4_nine × 13_nine = 53_nine. Give another calculation that would misinform an algebraic expression. Bonus: Why does 4_nine × 13_nine = 53_nine, or your calculation, give an incorrect idea for algebra?

Question 11

Calculate the sum and product of the pairs of polynomials.

A) $4 x ^ { 2 } + {}^{-} 7 x + 3 \text { and } 3 x + 5$

B) $5 x + {}^{-} 6 \text { and } {}^{-} 3 x + 7$

Question 12

Use specific values for m and n in $\left( a ^ { m } \right) ^ { n } = a ^ { m n }$
to give a basis for justifying that $a ^ { \frac { 1 } { 2}} = \sqrt { a }$
.

Question 13

Use $a ^ { m } \cdot a ^ { n } = a ^ { m + n }$ to justify defining $a ^ { \frac { 1 } { 2 } }$ = $\sqrt{a}$

Question 14

Use $\frac { a ^ { m } } { a ^ { n } } = a ^ { m - n }$ as the basis for defining a⁰ = 1 and a for nonzero values for a.

Question 15

Give the 100th and the nth entries for these lists, assuming the patterns continue. A) $12,22,32,42,52 , \ldots \quad$ $\quad$$\quad$ 100th $\underline{\quad\quad}$ $\quad$ nth $\underline{\quad\quad}$
B) $3,5,7,9,11 , \ldots \quad$ $\quad$$\quad$$\quad$$\quad$ 100th $\underline{\quad\quad}$ $\quad$ nth $\underline{\quad\quad}$
C) $2 \frac { 1 } { 2 } , 4,5 \frac { 1 } { 2 } , 7,8 \frac { 1 } { 2 } , 10 , \ldots \quad$ $\quad$ 100th $\underline{\quad\quad}$$\quad$$\quad$ nth $\underline{\quad\quad}$

Question 16

Give the first four numbers of each of the following sequences.

A) An arithmetic sequence with first term 19 and difference 20
B) A geometric sequence with first term 1.2 and ratio 3

Question 17

What is the difference between an arithmetic sequence and a geometric sequence?

Question 18

The repeating decimal for $\frac { 36350 } { 11111 }$ is 3.2715427154…. What digit is in the 99th decimal place in the repeating decimal? Explain how you know.

Question 19

Does each of the following give a function? Include correct reasons for credit.

A) Associate with each whole number n its third power n ³.
B) Assign to each person in the town his/her current last name.
C) Assign to the last names of people in town the first names.
D) The ordered pairs (5, 2), (7, 2), and (4, 9)
E) This dot diagram:

Question 20

Find a likely function rule for each of the following. Show your work.

A)
$\begin{array} { | l | l | } \hline x & f ( x ) \\\hline 2 & 15 \\\hline 3 & 19 \\\hline 4 & 23 \\\hline 5 & 27 \\\hline \ldots & \ldots \\\hline\end{array}$

f(x)=

B)
$\begin{array} { | l | l | } \hline x & y \\\hline 1 & 14 \\\hline 2 & 11 \\\hline 3 & 8 \\\hline 4 & 5 \\\hline \ldots & \ldots \\\hline\end{array}$

y=

C)
$\begin{array} { | l | l | } \hline x & g ( x ) \\\hline 4 & 29 \\\hline 3 & 21 \\\hline 1 & 5 \\\hline 2 & 13 \\\hline \ldots & \ldots \\\hline\end{array}$

g(x)=

D)
$\begin{array} { | l | l | } \hline x & y \\\hline 0 & 5 \\\hline 1 & 6 \\\hline 2 & 9 \\\hline 3 & 14 \\\hline 4 & 21 \\\hline \ldots & \ldots \\\hline\end{array}$

y=

Question 21

You find a function rule for a given table of data. Explain why your answer might not be the correct one.

Question 22

Two students have been looking for a function rule for the data below.
n | f(n)
1 3
2 5
3 8
4 12
Akeena: "I got f(n) =
n(n +1) + 2."
Bea: "Yes, but my mom worked a long time on it and said f(n) =
n(n + 1) + 2 + (n - 1)(n - 2)(n - 3)(n - 4). Let's ask the teacher."
How would you respond to the two students?

Question 23

A child is making "space modules with antennas" from toothpicks.
The child wonders, "How many toothpicks would it take to make a 100-room module with antennas?!"

Question 24

Take-home question: Here are examples of a shortcut for mentally squaring a number.
Example 1) 76²: Go to the closest multiple of 10-here, for 76, plus 4 to 80. Then go the opposite way from 76 by the same amount, minus 4 to 72. Multiply 80 and 72 mentally for 5760. Add the square of the up-down number 4. As a result, 5760 + 16 = 5776 and 76² = 5776.
Example 2) 62²: Go to 60 (down 2). Then go up from 62 by 2 to 64. Therefore, 60 × 64 = 3840. Add the square of 2 for 62² = 3844.
Example 3) 57²: Go to 60 then 54. Therefore, 60 × 54 = 3240. Add 3² for 57² = 3249.
Example 4) 198²: 200 × 196 = 39,200. Add 2² for 39,204 (= 198²).

A) Use the shortcut to calculate 37² mentally, and then write the mental steps you did.
B) Give a justification that the method works for squaring any n. Label the up-down number x.

Question 25

Finish each story problem so that it can be described by the given equation.

A) 50 - n = 16: "Jamal had 50 pieces of paper …"
B) 3n + 16.99 = 37: "Jose went shopping and bought a CD for $16.99 …"

Question 26

For each situation, write an equation using a variable. Then solve the equation to answer the question.

A) Krista has 4 more T-shirts than Leeann has, and Marika has 2 fewer T-shirts than Leeann has. The three students have 20 T-shirts in all. How many T-shirts does each have?
B) The entry fee at the fair is $5, and ride tickets cost 75¢ each. Pat spent $19.25 in all, for entry and ride tickets. How many ride tickets did Pat buy?

Question 27

Over a four-day period, one Girl Scout troop sold 178 boxes of cookies on the first day, 39 more boxes on the third day than on the second day, and 10 boxes fewer on the fourth day than on the second day. The troop sold 489 boxes in the four days. How many boxes did they sell each day?

Question 28

Suppose g(x) = 3x - 2, and h(x) is defined by machine X below. What number is each of the following? Show your work.

A) g(10) + g(5)
B) h(20)